The Compositional Structure of Bayesian Inference
This provides a theoretical foundation for structured statistical inference, potentially benefiting researchers in machine learning and statistics, though it appears incremental as it builds on existing categorical frameworks.
The paper tackles the problem of understanding the compositional structure of Bayesian inference, showing that inversion of complex processes can be computed piecewise and relating it to functional programming patterns, with results formulated in categorical terms for a type-driven approach.
Bayes' rule tells us how to invert a causal process in order to update our beliefs in light of new evidence. If the process is believed to have a complex compositional structure, we may observe that the inversion of the whole can be computed piecewise in terms of the component processes. We study the structure of this compositional rule, noting that it relates to the lens pattern in functional programming. Working in a suitably general axiomatic presentation of a category of Markov kernels, we see how we can think of Bayesian inversion as a particular instance of a state-dependent morphism in a fibred category. We discuss the compositional nature of this, formulated as a functor on the underlying category and explore how this can used for a more type-driven approach to statistical inference.